Problem: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{t^2 - 5t}{t^2 - 10t + 25}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - 5t}{t^2 - 10t + 25} = \dfrac{(t)(t - 5)}{(t - 5)(t - 5)} $ Notice that the term $(t - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 5)$ gives: $k = \dfrac{t}{t - 5}$ Since we divided by $(t - 5)$, $t \neq 5$. $k = \dfrac{t}{t - 5}; \space t \neq 5$